[FOM] Paolo Mancosu's monograph on 17th century mathematics
Martin Davis
martin at eipye.com
Mon Mar 24 17:17:20 EST 2003
I would like call the attention of FOMers to Paolo Mancosu's excellent
monograph PHILOSOPHY OF MATHEMATICS & MATHEMATICAL PRACTICE IN THE 17TH
CENTURY. I believe that by seeing how mathematicians grappled with problems
with infinity at this time, we can get valuable perspective on the
foundational issues of our day.
I found most striking the general reaction to a result by Torricelli (the
scientist who took over Galileo's chair in Florence). Torricelli computed
the finite volume enclosed by a particular infinite surface of revolution.
(In modern terms: the solid is formed by rotating about the X-axis the
rectangular hyperbola y = 1/x with x >= 1.) This result (today a homework
problem in a standard calculus course) created a sensation. I recommend to
all Mancosu's discussion of the way in which Torricelli's result played
havoc with the received wisdom of the day about the actual infinite.
I would like to suggest that mathematical practice (and especially the
formalisms which tend to give it much of its power) is by its nature
expansive, that concepts and methods tend to "overspill" leading to
FOUNDATIONAL questions about the status and validity of the new domains
hesitatingly revealed. The history of mathematics is replete with examples.
I believe that it is the work on extensions of ZFC, and especially on large
cardinals that is the main contemporary example of this phenomenon.
Martin
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
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http://www.eipye.com
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